Mathematics can be found everywhere. For our literary
analysis, we decided to look at the application of the friends and
strangers theory and Ramsey's Theory. These can be used to
understand the probability of the transplanted kidney surviving in
its new host. In our art analysis, we researched minimal surfaces
and how they are differentiated from the inefficient surfaces drawn
in the picture. We connected this to the way that the body
recognizes foreign tissue after a transplant. For our research
analysis, we connected the data to probability, and determined
possible outcomes of a patient based on Punnett squares and
heredity. Though kidneys may not appear to be an interesting part
of the body, mathematically they can be quite interesting. We hope
you enjoy reading our conclusions as much as we enjoyed developing
them.

- Graham DiNicola, Katie Robinson, Lia Sfiligoj

Mathematics in Literature

The New Kidney in Town by Mia Licciardi is a short narrative of
a kidney's adventure from a body that is dying in a car crash to
its transplant into the body of a young female child. It is not
until the kidney is actually transplanted that the adventure
begins. Is the kidney a "good enough" match to survive the
onslaught attack from the girl's natural immune system? It is the
"good enough" match that ultimately determines whether the girl
lives or dies.

How good the match has to be can be likened to a few different
math theories that deal with probability and likelihood of events
happening. The first theory is based on the math problem of
"Friends and Strangers." This classic problem begins by suggesting
that 6 people are randomly meeting. It can be assumed that no one
person is "friends" with all of the other five nor are they
"strangers" to all of the other five. If each person is friends
with some of the others and strangers to the remaining people, then
it follows that either 3 of these people were friends before they
entered the room or three people were complete strangers when they
entered the room. One of these two possibilities must exist. This
is often referred to as Ramsey's Theory. See Figure 1, where red
lines indicate that the people are strangers and the blue lines
indicate the people are friends. In this example, a red triangle
exists between C, E, and F; that means that those three people were
strangers to begin with. To remove this triangle - eliminating the
possibility that three people were strangers - would mean turning
one of the edges from red to blue. Any such change would create a
triangle of blue - meaning that there were three friends
beforehand. Changing line FC to blue creates a blue triangle ACF.
Changing either line EF or line CE to blue creates blue triangles
DEF and CDE respectively. Any coloring of these lines with only two
colors necessitates the creation of a triangle of one color or the
other.

In Figure 1, an example of Ramsey's Theory can be seen. "A"
represents the person who needs the kidney. The other letters, "B"
through "F," represent the other factors that affect whether the
kidney will function in the new body or not. These factors include
HIV, CMV, HCV and HSV and other genetic profiles, respectively. To
interpret this diagram, we assume that the blue line means that the
interaction does not affect the kidney negatively.

The only lines that matter under this interpretation are the
ones connected to "A." The ideal situation would be that all the
lines to "A" were blue - meaning none of the factors adversely
impacted the donor-kidney compatibility. But Ramsey's Theory rules
out this specific case. So, we consider other cases and imagine
that blue lines between characteristics (HIV, CMV, etc.) imply
supporting compatibility and red lines imply conflicting factors.
Thus, in our imagining of these circumstances, a blue triangle that
has "A" as a vertex is more desirable than two blue lines from "A"
that are connected with a red line. With this interpretation, we
can apply Ramsey's Theory.

If we inspect Figure 1, we see that there is no single triangle
that is composed of all blue lines. This means that the kidney is
not likely to be received by the body and will likely result in
rejection. One must keep in mind that Figure 1 only represents a
single combination that is possible for Ramsey's Theory. There are
approximately 30,000 different combinations that can exist for this
problem. This theory can have significance to the transplanting of
kidneys because as stated in the short story, a good quality
transplant is crucial to the child's survival. If the resulting
triangle
from the figure contains all red lines, then the match is not good
enough. If there exists a blue triangle, then this is a good match.
The likelihood of getting a blue triangle depends upon the initial
setup of the problem. As additional factors that affect the
rejection of a transplanted kidney are discovered, the Ramsey's
Theory can still be applied. The shape, as seen in Figure 1, is not
limited to only 6 factors. The number of interactions can be
increased and analyzed in a similar fashion.

As can be seen in Figure 2, there are many more kidney
transplants performed with cadaver kidneys than all other donors
combined.When a person receives a kidney from a cadaver, it would
be similar to the Ramsey's Theory, friends/strangers problem. There
is a small probability that a "perfect" match can be found,
although there is an increased probability that a "good enough"
match can be found. These "good enough" matches are transplanted,
but effective life of the kidney in the new body is the lowest when
compared to all other types of kidney

transplants, such as a transplant from a sibling or parent. As
can be seen in Figure 2, when a parent transplant occurs, the
kidney is expected to function properly in the new body for 12
years. It is only when there is an exact match of HLA-loci, which
is genetic matching, from a sibling, that the effective life of the
kidney increase significantly from a low of 9 years to a high of 22
years.

If Ramsey's Theory is applied to the data from Figure 2, and
groups of 6 were formed, then there would be 14,500 figures similar
to Figure 1 generated to represent all cadaver transplants. Each of
these figures would have the chance of either having two perfect
matches or two perfect mismatches. The combination of red and
blue-sided triangle would represent the possibility of having some
of the needed genetic matches but not enough to be considered a
"perfect" match. Ramsey's Theory helps portray why so many
transplants are performed using cadaver kidneys because of the
increased number of transplants that could be performed, while also
explaining why the survival rate is lower than the other types of
transplants.

The odds of a child's new kidney functioning properly after the
transplant are significantly lower than that of an adult's. Figure
3 shows the number of transplants performed each year for every age
group by The Kidney Transplant Program at Children's Hospital
Boston. There were only 43 out of the 623 transplants, 6.9%,
performed on children, up through age 10. This low number could be
the result of the low availability of kidneys for children. In the
story, the child was receiving a kidney from someone who had died
in a car accident. It was not apparent whether the kidney was also
from another child or she had received an adult kidney. However,
being a child there is a lower probability for a child to receive a
kidney transplant.

A second probability problem that can be applied to this short
story is the Monty Hall problem. Monty Hall hosted a game show,
called "Let's Make a Deal," where the contestant was given three
doors to choose from. For example, one door contained a car and the
other two contained goats. The contestant got to choose a door.
Before the door was opened, they were shown where one goat was by
opening a door, and then given the opportunity to switch to the
other door. Bayes' Theorem shows that there is an increased
probability of winning if in fact the contestant switches
doors.

The actual probability of winning when the contestant doesn't
change doors is 1/3.Whereas the probability of winning increases to
2/3 if the contestant switches their choice. Therefore one should
always switch to increase their probability of winning. This would
be a humorous twist if this were applied to the kidney transplant
problem. What if the young girl was given three kidneys to choose
from? One is a perfect match and the other two are not. Her doctor
asks her, "Which kidney do you want?" She chooses the kidney in
cooler #2. The doctor then opens cooler #3 and shows her that it is
not a perfect match. He then asks if she wants to switch her choice
to cooler #1. If the young girl doesn't change her choice, she has
a 1/3 chance of selecting the "perfect" kidney. If she switches her
choice, she increases her odds of selecting the "perfect" kidney to
2/3. Figure 5 verifies these odds, showing the results over 300
trials. Although a significant amount of scientific research has
been compiled to properly match donors and recipients of
transplants, simple probability still plays a role in deciding
whether the chosen organ will be effective or not.

Mathematics in Art

We chose the art piece titled Waiting on Fate by Janey
Eager. This piece depicts Pilot, an allegorical kidney, who is
mistaken by the creatures of the world Transplantia for a foreign
enemy. A central part of the piece is the bubble that Pilot is
staring at. A bubble is an example of a minimal space surface,
found often in nature. Nature, in the interest of conserving
energy, will select the shape that requires the least amount of
energy to maintain. Also, it can enclose a certain volume in as
little surface area as possible, another example of nature's desire
to conserve energy.

Joseph Antoine Ferdinand Plateau did a lot of work with minimal
surfaces. His research was geared towards finding the shape of
minimal surfaces. Today, it is known as Plateau's problem. Plateau
observed the way soap films clung to wires. The soap film fills its
boundary, the wire, in as little area as possible; therefore it is
a solution to Plateau's problem. For example, if a wire cube is
covered with soap film, which inverts in on itself, equalizing the
gases inside of it with the gases outside by introverting inside
the cube to enclose the wires in as little space as possible. The
pressure of the competing gases helps the bubble to form and
maintain its shape (see Figure 6). Without that pressure, the
formation of the cubic bubble would not be possible. Plateau also
did work with double bubbles and multiple bubbles, but the most
efficient form of a minimal space surface is the single bubble.

The bubble in the picture, therefore, is the only minimal space
surface in the drawing. Pilot's hair, clothing, and the various
machines and monsters surrounding him are very inefficient in terms
of the space of their surfaces. They cannot enclose as much space
that a sphere of the same surface
area could. The fact that the bubble is so starkly different than
the other creatures within the world of Transplantia sends a signal
to these creatures that Pilot is a foreign body. The artist's
interpretation can then be tied to Ms. Perez's research on
transplants. The body, much like the creatures, can recognize
foreign tissue that has been transplanted.

Towards the back of the picture is a spine-like object. The
spine contains central nerves that carry signals fromthe body to
the brain. The creatures of Transplantia, therefore, could
communicate with each other through this spine, and send signals to
each other of Pilot's presence. They believe that Pilot is a threat
to their town. This is similar to the body's ability to recognize
foreign tissue, and Pilot's fight to stay alive in Transplantia is
symbolic of the difficulties transplanted tissue undergoes in a new
host.

Mathematics In Research

Leat Perez's research deals with BK viruses, killer
immunoglobulin-like receptor genes in Chromosome 19, HLA glands,
and if these things influence infection with kidney transplants. In
laymen's terms, will the transplanted kidney work in the other
human's body? Before we get into the specifics, the background of
immunology is very important to understand when dealing with the
connection of math with Perez's research.

Immunology deals with how well one's body responds to
foreignness such as a new kidney. In order for a transplant to have
a chance to work, the Human Leukocyte Antigen (HLA) is matched. HLA
is a fancy way of describing proteins that contain a large amount
of genes associated with a specific person's immune functions. The
necessity to find the closest match of HLA is because there is a
chance that the patient's immune system will view the donor's HLA
as an invader and kill the kidney. The closer the match, the
greater the chances the kidney will survive.

Now where does our HLA come from? HLA are alleles, and a child
receives one from his mother and one from his father. With this
being said, one may predict the HLA of a couple's offspring using
Punnett squares to determine the recessive and dominant traits of
the offspring. Here are a few examples:

In Punnett squares, the genes of the parents are crossed to
predict the outcome of the offspring.In Punnett Square 1, 100% of
the offspring will have a dominant trait of HLA. This is because
the dominant trait prevails over the recessive, resulting in all
the offspring having a dominant HLA. As for when one crosses
dominant with recessive, the outcomes will be as follows: 25% of
offspring will be fully dominant, 50% will be dominant recessive,
and 25% will be recessive. One can conclude with Punnett Square 2
that 75% of the offspring will have a dominant HLA and 25% will
have a recessive HLA.

As Perez's research shows, killer immunoglobulin-like receptors
(KIR), which are categorized as natural killer cells, are
responsible for cleaning the body of foreign contaminants. Bottom
line: transplant + activated natural killer cells = organ
rejection. The KIR cells are found on Chromosome 19, and there are
only 4 characteristics:

Long Cytoplasmic Tail (LCT) vs.
Short Cytoplasmic Tail (SCT)

Two Domains (2D) vs. Three Domains
(3D)

Real Gene (RG) vs. Pseudo Gene
(PG)

Activating (AC) vs. Inhibitory
(IN)

These four combine to yield exactly 16 types of KIR cells.

The reason for only 16 possible types of KIR cells is the
mathematical expression of possibilities: 2n where n is 4 in this
case. The following are the 16 possible KIR cells without repeating
or doubling the selection:

As one may find, math is found in our everyday lives. Math helps
us understand the world better and offers a way to prove and back
up our findings. In Perez's research, math can be someone's best
possibility to survive - matching kidneys using math
expressions.